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Following https://thierrymoudiki.github.io/blog/2025/10/27/r/python/deterministic-shift-caps-swaptions, I propose three distinct methods for short rate construction—ranging from parametric (Nelson-Siegel) to fully data-driven approaches—and derive a deterministic shift adjustment ensuring consistency with the fundamental theorem of asset pricing. The framework naturally integrates with modern statistical learning methods, including conformal prediction and copula-based forecasting. Numerical experiments demon- strate accurate calibration to market zero-coupon bond prices and reliable pricing of interest rate derivatives including caps and swaptions
I developed in conjunction with these short rates models, a flexible framework for arbitrage-free simulation of short rates that reconciles descriptive yield curve models with no-arbitrage pricing theory. Unlike existing approaches that require strong parametric assumptions, the method accommodates any bounded, continuous, and simulable short rate process.
In this post, we implement three methods for constructing instantaneous short rates from historical yield curves, as described in the preprint https://www.researchgate.net/publication/393794192_New_Short_Rate_Models_and_their_Arbitrage-Free_Extension_A_Flexible_Framework_for_Historical_and_Market-Consistent_Simulation. We also implement the deterministic shift adjustment to ensure arbitrage-free pricing of caps and swaptions.
install.packages(c('randomForest', 'NMOF'))
install.packages('gridExtra')
library(tidyverse)
library(randomForest)
library(NMOF) # For NS() and NSS() functions
set.seed(123)
# ============================================================================
# 1. Generate Synthetic Yield Curve Data (Vectorized with NMOF)
# ============================================================================
generate_yield_curve_ns <- function(n_dates = 100,
maturities = c(0.25, 0.5, 1, 2, 3, 5, 7, 10)) {
dates <- 1:n_dates
# Time-varying Nelson-Siegel factors
beta1 <- 0.06 + 0.01 * sin(2 * pi * dates / 50) + rnorm(n_dates, 0, 0.002)
beta2 <- -0.02 + cumsum(rnorm(n_dates, 0, 0.005)) / sqrt(dates)
beta3 <- 0.01 + rnorm(n_dates, 0, 0.003)
lambda <- 0.0609
# Vectorized NS yield generation using NMOF::NS
# NS(param, tm) where param = c(beta1, beta2, beta3, lambda)
yields <- matrix(0, n_dates, length(maturities))
for (i in 1:n_dates) {
yields[i, ] <- NS(param = c(beta1[i], beta2[i], beta3[i], lambda),
tm = maturities)
}
list(yields = yields, maturities = maturities,
beta1 = beta1, beta2 = beta2, beta3 = beta3, lambda = lambda,
type = "NS")
}
generate_yield_curve_nss <- function(n_dates = 100,
maturities = c(0.25, 0.5, 1, 2, 3, 5, 7, 10)) {
dates <- 1:n_dates
# Time-varying Nelson-Siegel-Svensson factors
beta1 <- 0.06 + 0.01 * sin(2 * pi * dates / 50) + rnorm(n_dates, 0, 0.002)
beta2 <- -0.02 + cumsum(rnorm(n_dates, 0, 0.005)) / sqrt(dates)
beta3 <- 0.01 + rnorm(n_dates, 0, 0.003)
beta4 <- 0.005 + rnorm(n_dates, 0, 0.002) # Additional NSS factor
lambda1 <- 0.0609
lambda2 <- 0.50 # Second decay parameter for NSS
# Vectorized NSS yield generation using NMOF::NSS
# NSS(param, tm) where param = c(beta1, beta2, beta3, beta4, lambda1, lambda2)
yields <- matrix(0, n_dates, length(maturities))
for (i in 1:n_dates) {
yields[i, ] <- NSS(param = c(beta1[i], beta2[i], beta3[i], beta4[i],
lambda1, lambda2),
tm = maturities)
}
list(yields = yields, maturities = maturities,
beta1 = beta1, beta2 = beta2, beta3 = beta3, beta4 = beta4,
lambda1 = lambda1, lambda2 = lambda2,
type = "NSS")
}
# ============================================================================
# 2. Method 1: NS/NSS Extrapolation to Zero Maturity (Analytical Limit)
# ============================================================================
method1_ns_extrapolation <- function(data) {
if (data$type == "NS") {
# r(t) = lim_{tau->0} NS(tau) = beta1 + beta2
# Since lim_{tau->0} (1-exp(-tau/lambda))/(tau/lambda) = 1
# and lim_{tau->0} exp(-tau/lambda) = 1
short_rates <- data$beta1 + data$beta2
} else if (data$type == "NSS") {
# r(t) = lim_{tau->0} NSS(tau) = beta1 + beta2 + beta4
# Both slope terms converge to 1 as tau->0
short_rates <- data$beta1 + data$beta2 + data$beta4
}
short_rates
}
# ============================================================================
# 3. Method 2: NS/NSS Features with Random Forest (Using Small Tau)
# ============================================================================
method2_ns_ml <- function(data) {
n_dates <- nrow(data$yields)
short_rates <- numeric(n_dates)
# Use very small tau to approximate zero (e.g., 1 day = 1/252 years)
tau_tiny <- 1/252
if (data$type == "NS") {
for (i in 1:n_dates) {
# Create NS features using NMOF::NSf (factor loadings)
# NSf(lambda, tm) returns matrix: rows = maturities, cols = 3 factors
loadings_matrix <- NSf(lambda = data$lambda, tm = data$maturities)
# Convert to data frame
features <- as.data.frame(loadings_matrix)
colnames(features) <- c("L1", "L2", "L3")
target <- data$yields[i, ]
# Train RF
rf <- randomForest(x = features, y = target, ntree = 50, nodesize = 2)
# Predict at tau ≈ 0: Use analytical limit values (1, 1, 0)
# OR use very small tau
# Analytical approach (more accurate):
features_zero <- data.frame(L1 = 1, L2 = 1, L3 = 0)
short_rates[i] <- suppressWarnings(predict(rf, features_zero))
}
} else if (data$type == "NSS") {
for (i in 1:n_dates) {
# Create NSS features using NMOF::NSSf
# NSSf(lambda1, lambda2, tm) returns matrix: rows = maturities, cols = 4 factors
loadings_matrix <- NSSf(lambda1 = data$lambda1,
lambda2 = data$lambda2,
tm = data$maturities)
# Convert to data frame
features <- as.data.frame(loadings_matrix)
colnames(features) <- c("L1", "L2", "L3", "L4")
target <- data$yields[i, ]
# Train RF
rf <- randomForest(x = features, y = target, ntree = 50, nodesize = 2)
# Predict at tau ≈ 0: Use analytical limit values (1, 1, 0, 0)
# Note: L4 = (1-exp(-aux2))/aux2 - exp(-aux2) -> 1 - 1 = 0 as tau->0
features_zero <- data.frame(L1 = 1, L2 = 1, L3 = 0, L4 = 0)
short_rates[i] <- suppressWarnings(predict(rf, features_zero))
}
}
short_rates
}
# ============================================================================
# 4. Method 3: Direct Linear Regression to Zero Maturity
# ============================================================================
method3_direct_regression <- function(data) {
n_dates <- nrow(data$yields)
short_rates <- numeric(n_dates)
for (i in 1:n_dates) {
# Fit linear model: R(tau) = a + b*tau
model <- lm(data$yields[i, ] ~ data$maturities)
# Extrapolate to tau=0
short_rates[i] <- coef(model)[1]
}
short_rates
}
# ============================================================================
# 5. Simulate Short Rate Paths
# ============================================================================
simulate_paths <- function(short_rates, n_sims = 100,
horizon = 10, dt = 0.25) {
n_steps <- horizon / dt
paths <- matrix(0, n_sims, n_steps + 1)
# AR(1) model
r0 <- tail(short_rates, 1)
mu <- mean(short_rates)
phi <- 0.95
sigma <- sd(diff(short_rates)) * sqrt(dt)
paths[, 1] <- r0
for (t in 2:(n_steps + 1)) {
paths[, t] <- mu + phi * (paths[, t-1] - mu) + rnorm(n_sims, 0, sigma)
paths[, t] <- pmax(paths[, t], 0.001)
}
paths
}
# ============================================================================
# 6. Compute Unadjusted ZCB Prices (Vectorized)
# ============================================================================
compute_unadjusted_zcb <- function(paths, maturities, dt = 0.25) {
n_sims <- nrow(paths)
zcb_prices <- numeric(length(maturities))
for (i in 1:length(maturities)) {
T_mat <- maturities[i]
n_steps <- round(T_mat / dt)
# Vectorized integration
integrals <- rowSums(paths[, 1:n_steps, drop = FALSE]) * dt
zcb_prices[i] <- mean(exp(-integrals))
}
zcb_prices
}
# ============================================================================
# 7. Market ZCB Prices
# ============================================================================
get_market_zcb <- function(data, date_idx, target_maturities) {
yields <- data$yields[date_idx, ]
maturities <- data$maturities
# Interpolate yields
yields_interp <- approx(maturities, yields, target_maturities)$y
return(exp(-yields_interp * target_maturities))
}
# ============================================================================
# 8. Deterministic Shift Adjustment (Vectorized)
# ============================================================================
apply_deterministic_shift <- function(paths, zcb_unadjusted, zcb_market,
maturities, dt = 0.25) {
n_sims <- nrow(paths)
# Compute simulated forward rates (vectorized)
f_sim <- numeric(length(maturities))
for (i in 1:length(maturities)) {
T_mat <- maturities[i]
n_steps <- round(T_mat / dt)
r_T <- paths[, n_steps]
integrals <- rowSums(paths[, 1:n_steps, drop = FALSE]) * dt
discount <- exp(-integrals)
f_sim[i] <- sum(r_T * discount) / sum(discount)
}
# Market forward rates
f_market <- -diff(log(zcb_market)) / diff(maturities)
f_market <- c(f_market[1], f_market)
# Shift function
phi <- f_market - f_sim
# Adjusted ZCB prices (vectorized)
zcb_adjusted <- numeric(length(maturities))
for (i in 1:length(maturities)) {
if (i == 1) {
int_phi <- phi[1] * maturities[1]
} else {
int_phi <- sum(phi[1:i] * diff(c(0, maturities[1:i])))
}
zcb_adjusted[i] <- exp(-int_phi) * zcb_unadjusted[i]
}
return(list(zcb_adjusted = zcb_adjusted,
phi = phi))
}
# ============================================================================
# 9. Calibrated Confidence Intervals for Adjusted Prices
# ============================================================================
compute_adjusted_confidence_intervals <- function(paths, zcb_market, maturities,
n_boot = 500, dt = 0.25,
alpha = 0.01) {
n_sims <- nrow(paths)
boot_adjusted_prices <- matrix(0, n_boot, length(maturities))
for (b in 1:n_boot) {
idx <- sample(1:n_sims, n_sims, replace = TRUE)
boot_paths <- paths[idx, ]
boot_unadj <- compute_unadjusted_zcb(boot_paths, maturities, dt)
boot_adjustment <- apply_deterministic_shift(boot_paths, boot_unadj,
zcb_market, maturities, dt)
boot_adjusted_prices[b, ] <- boot_adjustment$zcb_adjusted
}
ci_lower <- apply(boot_adjusted_prices, 2, quantile, probs = alpha/2)
ci_upper <- apply(boot_adjusted_prices, 2, quantile, probs = 1 - alpha/2)
return(list(lower = ci_lower, upper = ci_upper))
}
# ============================================================================
# 10. Run Analysis Function
# ============================================================================
run_analysis <- function(data, model_name) {
cat(sprintf("\n========== %s ANALYSIS ==========\n\n", model_name))
test_maturities <- c(1, 3, 5, 7, 10)
zcb_market_test <- get_market_zcb(data, nrow(data$yields), test_maturities)
cat("Market ZCB Prices:\n")
print(data.frame(Maturity = test_maturities, Price = round(zcb_market_test, 6)))
cat("\n")
results_list <- list()
# Method 1: NS/NSS Extrapolation
cat("--- METHOD 1: Short rates \n with NS/NSS Extrapolation ---\n")
r1 <- method1_ns_extrapolation(data)
cat(sprintf("Short rate (last obs): %.4f%%\n", r1[length(r1)] * 100))
paths1 <- simulate_paths(r1, n_sims = 100L)
zcb_unadj1 <- compute_unadjusted_zcb(paths1, test_maturities)
adjustment1 <- apply_deterministic_shift(paths1, zcb_unadj1, zcb_market_test,
test_maturities)
cat("Computing calibrated confidence intervals...\n")
ci1 <- compute_adjusted_confidence_intervals(paths1, zcb_market_test,
test_maturities, n_boot = 500)
results1 <- data.frame(
Maturity = test_maturities,
Market = round(zcb_market_test, 6),
Unadjusted = round(zcb_unadj1, 6),
Adjusted = round(adjustment1$zcb_adjusted, 6),
Error_pct = round(abs(adjustment1$zcb_adjusted - zcb_market_test) /
zcb_market_test * 100, 3),
CI_Lower = round(ci1$lower, 6),
CI_Upper = round(ci1$upper, 6),
CI_Width_bps = round((ci1$upper - ci1$lower) * 10000, 1),
Market_In_CI = zcb_market_test >= ci1$lower & zcb_market_test <= ci1$upper
)
cat("\nResults:\n")
print(results1)
cat("\n")
results_list[[1]] <- results1 %>% mutate(Method = "Method 1: NS/NSS Extrapolation")
# Method 2: NS/NSS + Random Forest
cat("--- METHOD 2: Short rates \n with NS/NSS + Random Forest ---\n")
r2 <- method2_ns_ml(data)
cat(sprintf("Short rate (last obs): %.4f%%\n", r2[length(r2)] * 100))
paths2 <- simulate_paths(r2, n_sims = 100L)
zcb_unadj2 <- compute_unadjusted_zcb(paths2, test_maturities)
adjustment2 <- apply_deterministic_shift(paths2, zcb_unadj2, zcb_market_test,
test_maturities)
cat("Computing calibrated confidence intervals...\n")
ci2 <- compute_adjusted_confidence_intervals(paths2, zcb_market_test,
test_maturities, n_boot = 500)
results2 <- data.frame(
Maturity = test_maturities,
Market = round(zcb_market_test, 6),
Unadjusted = round(zcb_unadj2, 6),
Adjusted = round(adjustment2$zcb_adjusted, 6),
Error_pct = round(abs(adjustment2$zcb_adjusted - zcb_market_test) /
zcb_market_test * 100, 3),
CI_Lower = round(ci2$lower, 6),
CI_Upper = round(ci2$upper, 6),
CI_Width_bps = round((ci2$upper - ci2$lower) * 10000, 1),
Market_In_CI = zcb_market_test >= ci2$lower & zcb_market_test <= ci2$upper
)
cat("\nResults:\n")
print(results2)
cat("\n")
results_list[[2]] <- results2 %>% mutate(Method = "Method 2: NS/NSS + RF")
# Method 3: Direct Regression
cat("--- METHOD 3: Short rates \n with Direct Regression ---\n")
r3 <- method3_direct_regression(data)
cat(sprintf("Short rate (last obs): %.4f%%\n", r3[length(r3)] * 100))
paths3 <- simulate_paths(r3, n_sims = 100L)
zcb_unadj3 <- compute_unadjusted_zcb(paths3, test_maturities)
adjustment3 <- apply_deterministic_shift(paths3, zcb_unadj3, zcb_market_test,
test_maturities)
cat("Computing calibrated confidence intervals...\n")
ci3 <- compute_adjusted_confidence_intervals(paths3, zcb_market_test,
test_maturities, n_boot = 500)
results3 <- data.frame(
Maturity = test_maturities,
Market = round(zcb_market_test, 6),
Unadjusted = round(zcb_unadj3, 6),
Adjusted = round(adjustment3$zcb_adjusted, 6),
Error_pct = round(abs(adjustment3$zcb_adjusted - zcb_market_test) /
zcb_market_test * 100, 3),
CI_Lower = round(ci3$lower, 6),
CI_Upper = round(ci3$upper, 6),
CI_Width_bps = round((ci3$upper - ci3$lower) * 10000, 1),
Market_In_CI = zcb_market_test >= ci3$lower & zcb_market_test <= ci3$upper
)
cat("\nResults:\n")
print(results3)
cat("\n")
results_list[[3]] <- results3 %>% mutate(Method = "Method 3: Direct Regression")
# Summary
cat("=== SUMMARY ===\n")
cat(sprintf("Method 1 - Mean absolute error: %.3f%%\n", mean(results1$Error_pct)))
cat(sprintf("Method 2 - Mean absolute error: %.3f%%\n", mean(results2$Error_pct)))
cat(sprintf("Method 3 - Mean absolute error: %.3f%%\n", mean(results3$Error_pct)))
coverage1 <- mean(results1$Market_In_CI) * 100
coverage2 <- mean(results2$Market_In_CI) * 100
coverage3 <- mean(results3$Market_In_CI) * 100
cat("\n=== CI Coverage ===\n")
cat(sprintf("Method 1 - Market in 99%% CI: %.0f%%\n", coverage1))
cat(sprintf("Method 2 - Market in 99%% CI: %.0f%%\n", coverage2))
cat(sprintf("Method 3 - Market in 99%% CI: %.0f%%\n", coverage3))
cat(sprintf("\nAvg CI Width: %.1f bps (M1), %.1f bps (M2), %.1f bps (M3)\n",
mean(results1$CI_Width_bps), mean(results2$CI_Width_bps),
mean(results3$CI_Width_bps)))
return(bind_rows(results_list))
}
# ============================================================================
# 11. Main Execution
# ============================================================================
cat("=== Arbitrage-Free Framework: NMOF Implementation ===\n")
cat("Using vectorized NS() and NSS() functions\n\n")
# Generate data for both NS and NSS
cat("Generating synthetic yield curves...\n")
data_ns <- generate_yield_curve_ns(n_dates = 100)
data_nss <- generate_yield_curve_nss(n_dates = 100)
# Run analysis for NS
results_ns <- run_analysis(data_ns, "NELSON-SIEGEL (NS)")
# Run analysis for NSS
results_nss <- run_analysis(data_nss, "NELSON-SIEGEL-SVENSSON (NSS)")
# ============================================================================
# 12. Comparative Visualization
# ============================================================================
cat("\n=== Generating Comparative Visualization ===\n")
library(gridExtra)
# Add model type to results
results_ns <- results_ns %>% mutate(Model = "NS")
results_nss <- results_nss %>% mutate(Model = "NSS")
all_results <- bind_rows(results_ns, results_nss)
# Plot 3: Adjusted vs Market Prices
p3 <- ggplot(all_results, aes(x = Maturity)) +
geom_point(aes(y = Market), size = 3, shape = 4, stroke = 2) +
geom_point(aes(y = Adjusted, color = Model), size = 2, alpha = 0.7) +
geom_errorbar(aes(ymin = CI_Lower, ymax = CI_Upper, color = Model),
width = 0.2, alpha = 0.5) +
facet_wrap(~Method, ncol = 3) +
scale_color_manual(values = c("NS" = "#E41A1C", "NSS" = "#377EB8")) +
labs(title = "Adjusted Prices with Confidence Intervals",
subtitle = "Market prices (×) vs Adjusted prices (•) with 95% CI",
x = "Maturity (Years)", y = "Zero-Coupon Bond Prices") +
theme_minimal(base_size = 11) +
theme(legend.position = "bottom", strip.text = element_text(face = "bold"))
print(p3)
=== Arbitrage-Free Framework: NMOF Implementation ===
Using vectorized NS() and NSS() functions
Generating synthetic yield curves...
========== NELSON-SIEGEL (NS) ANALYSIS ==========
Market ZCB Prices:
Maturity Price
1 1 0.944368
2 3 0.841025
3 5 0.748991
4 7 0.667029
5 10 0.560591
--- METHOD 1: Short rates
with NS/NSS Extrapolation ---
Short rate (last obs): 3.2570%
Computing calibrated confidence intervals...
Results:
Maturity Market Unadjusted Adjusted Error_pct CI_Lower CI_Upper
1 1 0.944368 0.967691 0.943862 0.054 0.943584 0.944152
2 3 0.841025 0.905163 0.841066 0.005 0.840352 0.841831
3 5 0.748991 0.846230 0.749097 0.014 0.748257 0.750021
4 7 0.667029 0.791175 0.667194 0.025 0.666362 0.668134
5 10 0.560591 0.715174 0.560692 0.018 0.559592 0.561671
CI_Width_bps Market_In_CI
1 5.7 FALSE
2 14.8 TRUE
3 17.6 TRUE
4 17.7 TRUE
5 20.8 TRUE
--- METHOD 2: Short rates
with NS/NSS + Random Forest ---
Short rate (last obs): 5.6769%
Computing calibrated confidence intervals...
Results:
Maturity Market Unadjusted Adjusted Error_pct CI_Lower CI_Upper
1 1 0.944368 0.944570 0.943958 0.043 0.943678 0.944300
2 3 0.841025 0.841244 0.841482 0.054 0.840561 0.842174
3 5 0.748991 0.748978 0.749251 0.035 0.748509 0.749930
4 7 0.667029 0.666754 0.667387 0.054 0.666451 0.668276
5 10 0.560591 0.558617 0.561682 0.195 0.560760 0.562701
CI_Width_bps Market_In_CI
1 6.2 FALSE
2 16.1 TRUE
3 14.2 TRUE
4 18.3 TRUE
5 19.4 FALSE
--- METHOD 3: Short rates
with Direct Regression ---
Short rate (last obs): 5.6515%
Computing calibrated confidence intervals...
Results:
Maturity Market Unadjusted Adjusted Error_pct CI_Lower CI_Upper
1 1 0.944368 0.944990 0.943743 0.066 0.943456 0.944000
2 3 0.841025 0.843021 0.841146 0.014 0.840457 0.841841
3 5 0.748991 0.751161 0.749289 0.040 0.748411 0.750119
4 7 0.667029 0.668924 0.667445 0.062 0.666569 0.668325
5 10 0.560591 0.561686 0.560619 0.005 0.559444 0.561621
CI_Width_bps Market_In_CI
1 5.4 FALSE
2 13.8 TRUE
3 17.1 TRUE
4 17.6 TRUE
5 21.8 TRUE
=== SUMMARY ===
Method 1 - Mean absolute error: 0.023%
Method 2 - Mean absolute error: 0.076%
Method 3 - Mean absolute error: 0.037%
=== CI Coverage ===
Method 1 - Market in 99% CI: 80%
Method 2 - Market in 99% CI: 60%
Method 3 - Market in 99% CI: 80%
Avg CI Width: 15.3 bps (M1), 14.8 bps (M2), 15.1 bps (M3)
========== NELSON-SIEGEL-SVENSSON (NSS) ANALYSIS ==========
Market ZCB Prices:
Maturity Price
1 1 0.941887
2 3 0.836276
3 5 0.742752
4 7 0.659696
5 10 0.552198
--- METHOD 1: Short rates
with NS/NSS Extrapolation ---
Short rate (last obs): 4.6355%
Computing calibrated confidence intervals...
Results:
Maturity Market Unadjusted Adjusted Error_pct CI_Lower CI_Upper
1 1 0.941887 0.954688 0.942270 0.041 0.941853 0.942716
2 3 0.836276 0.871008 0.835715 0.067 0.834544 0.836739
3 5 0.742752 0.794980 0.741836 0.123 0.740349 0.743138
4 7 0.659696 0.726242 0.658950 0.113 0.657747 0.660143
5 10 0.552198 0.633806 0.551628 0.103 0.549925 0.553257
CI_Width_bps Market_In_CI
1 8.6 TRUE
2 21.9 TRUE
3 27.9 TRUE
4 24.0 TRUE
5 33.3 TRUE
--- METHOD 2: Short rates
with NS/NSS + Random Forest ---
Short rate (last obs): 5.9567%
Computing calibrated confidence intervals...
Results:
Maturity Market Unadjusted Adjusted Error_pct CI_Lower CI_Upper
1 1 0.941887 0.942313 0.942164 0.029 0.941784 0.942523
2 3 0.836276 0.836567 0.836628 0.042 0.835661 0.837590
3 5 0.742752 0.742821 0.743231 0.064 0.741899 0.744219
4 7 0.659696 0.658894 0.660377 0.103 0.659165 0.661587
5 10 0.552198 0.549714 0.552640 0.080 0.551440 0.554016
CI_Width_bps Market_In_CI
1 7.4 TRUE
2 19.3 TRUE
3 23.2 TRUE
4 24.2 TRUE
5 25.8 TRUE
--- METHOD 3: Short rates
with Direct Regression ---
Short rate (last obs): 5.9770%
Computing calibrated confidence intervals...
Results:
Maturity Market Unadjusted Adjusted Error_pct CI_Lower CI_Upper
1 1 0.941887 0.942102 0.942063 0.019 0.941738 0.942405
2 3 0.836276 0.836988 0.836231 0.005 0.835452 0.837112
3 5 0.742752 0.742843 0.743059 0.041 0.742083 0.744069
4 7 0.659696 0.658711 0.659669 0.004 0.658707 0.660866
5 10 0.552198 0.550538 0.552046 0.028 0.550710 0.553238
CI_Width_bps Market_In_CI
1 6.7 TRUE
2 16.6 TRUE
3 19.9 TRUE
4 21.6 TRUE
5 25.3 TRUE
=== SUMMARY ===
Method 1 - Mean absolute error: 0.089%
Method 2 - Mean absolute error: 0.064%
Method 3 - Mean absolute error: 0.019%
=== CI Coverage ===
Method 1 - Market in 99% CI: 100%
Method 2 - Market in 99% CI: 100%
Method 3 - Market in 99% CI: 100%
Avg CI Width: 23.1 bps (M1), 20.0 bps (M2), 18.0 bps (M3)
=== Generating Comparative Visualization ===
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